An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

A new theory of space syntax

Relations between different components of urban structure are often measured in aliteral manner, along streets for example, the usual representation being routesbetween junctions which form the nodes of an equivalent planar graph. A popularvariant on this theme ? space syntax ? treats these routes as streets containing one ormore junctions, with the equivalent graph representation being more abstract, basedon relations between the streets which themselves are treated as nodes. In this paper,we articulate space syntax as a specific case of relations between any two sets, in thiscase, streets and their junctions, from which we derive two related representations.The first or primal problem is traditional space syntax based on relations betweenstreets through their junctions; the second or dual problem is the more usualmorphological representation of relations between junctions through their streets.The unifying framework that we propose suggests we shift our focus from the primalproblem where accessibility or distance is associated with lines or streets, to the dualproblem where accessibility is associated with points or junctions. This traditionalrepresentation of accessibility between points rather than between lines is easier tounderstand and makes more sense visually. Our unifying framework enables us toeasily shift from the primal problem to the dual and back, thus providing a muchricher interpretation of the syntax. We develop an appropriate algebra which providesa clearer approach to connectivity and distance in the equivalent graphrepresentations, and we then demonstrate these variants for the primal and dualproblems in one of the first space syntax street network examples, the French villageof Gassin. An immediate consequence of our analysis is that we show how the directconnectivity of streets (or junctions) to one another is highly correlated with thedistance measures used. This suggests that a simplified form of syntax can beoperationalized through counts of streets and junctions in the original street network

Logical limits of abstract argumentation frameworks

International audienceDung’s (1995) argumentation framework takes as input two abstract entities: a set of arguments and a binary relation encoding attacks between these arguments. It returns acceptable sets of arguments, called extensions, w.r.t. a given semantics. While the abstract nature of this setting is seen as a great advantage, it induces a big gap with the application that it is used to. This raises some questions about the compatibility of the setting with a logical formalism (i.e., whether it is possible to instantiate it properly from a logical knowledge base), and about the significance of the various semantics in the application context. In this paper we tackle the above questions. We first propose to fill in the previous gap by extending Dung’s (1995) framework. The idea is to consider all the ingredients involved in an argumentation process. We start with the notion of an abstract monotonic logic which consists of a language (defining the formulas) and a consequence operator. We show how to build, in a systematic way, arguments from a knowledge base formalised in such a logic. We then recall some basic postulates that any instantiation should satisfy. We study how to choose an attack relation so that the instantiation satisfies the postulates. We show that symmetric attack relations are generally not suitable. However, we identify at least one ‘appropriate’ attack relation. Next, we investigate under stable, semi-stable, preferred, grounded and ideal semantics the outputs of logic-based instantiations that satisfy the postulates. For each semantics, we delimit the number of extensions an argumentation system may have, characterise the extensions in terms of subsets of the knowledge base, and finally characterise the set of conclusions that are drawn from the knowledge base. The study reveals that stable, semi-stable and preferred semantics either lead to counter-intuitive results or provide no added value w.r.t. naive semantics. Besides, naive semantics either leads to arbitrary results or generalises the coherence-based approach initially developed by Rescher and Manor (1970). Ideal and grounded semantics either coincide and generalise the free consequence relation developed by Benferhat, Dubois, and Prade (1997), or return arbitrary results. Consequently, Dung’s (1995) framework seems problematic when applied over deductive logical formalisms

Scalable Scheduling of Energy Control Systems

Peak power consumption is a universal problem across energy control systems in electrical grids, buildings, and industrial automation where the uncoordinated operation of multiple controllers result in temporally correlated electricity demand surges (or peaks). While there exist several different approaches to balance power consumption by load shifting and load shedding, they operate on coarse grained time scales and do not help in de-correlating energy sinks. The Energy System Scheduling Problem is particularly hard due to its binary control variables. Its complexity grows exponentially with the scale of the system, making it impossible to handle systems with more than a few variables. We developed a scalable approach for fine-grained scheduling of energy control systems that novelly combines techniques from control theory and computer science. The original system with binary control variables are approximated by an averaged system whose inputs are the utilization values of the binary inputs within a given period. The error between the two systems can be bounded, which allows us to derive a safety constraint for the averaged system so that the original system\u27s safety is guaranteed. To further reduce the complexity of the scheduling problem, we abstract the averaged system by a simple single-state single-input dynamical system whose control input is the upper-bound of the total demand of the system. This model abstraction is achieved by extending the concept of simulation relations between transition systems to allow for input constraints between the systems. We developed conditions to test for simulation relations as well as algorithms to compute such a model abstraction. As a consequence, we only need to solve a small linear program to compute an optimal bound of the total demand. The total demand is then broken down, by solving a linear program much smaller than the original program, to individual utilization values of the subsystems, whose actual schedule is then obtained by a low-level scheduling algorithm. Numerical simulations in Matlab show the effectiveness and scalability of our approach

An order-theoretic analysis of interpretations among propositional deductive systems

In this paper we study interpretations and equivalences of propositional deductive systems by using a quantale-theoretic approach introduced by Galatos and Tsinakis. Our aim is to provide a general order-theoretic framework which is able to describe and characterize both strong and weak forms of interpretations among propositional deductive systems also in the cases where the systems have different underlying languages